As a substituted Andy Carroll stomped straight off down the tunnel muttering a few choice expletives while neglecting to shake his manager’s hand believers in Sabermetrics must have rubbed their hands in glee.

The principle of using statistics to find value in unlikely places, Sabermetrics was born in baseball and later underpinned the famous Moneyball concept supposedly beloved of the Anfield hierarchy.

Moneyball devotees claim that maximising the untapped potential of previously undervalued players is the key to sporting success but it is Newcastle United who appear to be implementing the idea rather better than Liverpool.

Despite spending a mere £26m during 14 months which have seen Kenny Dalglish part with £115m, Alan Pardew’s side stand sixth, 11 points ahead of eighth-placed Liverpool and kept behind Chelsea only by goal difference.

On a day when Dalglish presided over his team’s sixth defeat in the past seven Premier League games and José Enrique played the last seven minutes in the away goal after Pepe Reina was sent off for an attempted headbutt on James Perch, the £35m the Anfield club paid Newcastle for Carroll seemed more inexplicable than ever.

What was crystal clear, though, is that while Carroll has regressed, Pardew’s side have moved on, something emphasised by two more goals from Papiss Cissé.

Dalglish was never overly popular during a short stint as Newcastle manager during the mid 1990s and the home crowd quickly delighted in serenading him with a cheeky song. “Sacked in the morning, you’re getting sacked in the morning,” they chorused in the wake of Cissé’s opener, a diving header which flew past Reina before going in off the opposite upright. It was Cissé’s sixth goal in six games. Yet even the Senegal striker would acknowledge that it would never have happened without Hatem Ben Arfa’s brilliance.

The duel between Ben Arfa, operating on the right of Alan Pardew’s front three, and Enrique, Liveprool’s former Newcastle left-back, had already turned into a compelling sub-plot by the time the Frenchman’s skill left first Jonjo Shelvey and then Enrique confounded. Momentarily free of minders, Ben Arfa curled a sublime, goal-creating, cross towards the far post where Cissé comfortably out-jumped Martin Skrtel.

Until then Liverpool had looked the team more likely to score. They probably would have done if Tim Krul had not reacted so adroitly in tipping Craig Bellamy’s deflected cross onto the bar or the returning Carroll stayed on his feet instead of collecting a booking for simulation following an apparent dive after rounding Newcastle’s goalkeeper.

Having beaten Mike Williamson in the air before surging past the home centre-half, Carroll found himself with only Krul to beat. Although, by spreading himself, the Dutchman forced his erstwhile team-mate wide, Carroll seemed to have succeeded in circumventing him before collapsing in an ungainly heap. As Liverpool appealed for a penalty, Krul, who had not touched the centre-forward, reacted furiously, his anger only abating when the referee Martin Atkinson waved a yellow card in Carroll’s face.

After that little cameo Carroll was treated to a couple of painful, undetected little ankle taps from Yohan Cabaye and Danny Guthrie. Indeed when Cissé scored the visiting centre-forward was receiving treatment by the technical area and briefly looked close to tears.

Fortunately for Dalglish, Bellamy, another member of Liverpool’s Gallowgate old boys’ contingent, is made of sterner stuff and continued to delight in getting behind José Gutiérrez, Pardew’s makeshift left-back.

For once, though, even Bellamy found himself eclipsed. As half time drew nearer and Newcastle grew stronger, all eyes were drawn to Ben Arfa and his repeated humiliations of Enrique.

If it helped that the body language of certain Liverpool players – Luis Suárez emitted an unbecoming wail of anguish when Steven Gerrard uncharacteristically fluffed a through pass which would have left the onside Uruguayan clean through – gave away signs of their frustration, the Newcastle manager really could have done with Williamson opening the second period by scoring rather than heading against a post.

By now Dalglish had ordered Bellamy to play through the middle with Carroll to his left and Suárez on the right. Although Carroll appeared lost, Bellamy menaced and would surely have scored had Gerrard not fractionally overhit a cross, thereby permitting Krul to gather as the Welshman zoomed towards him.

Dalglish’s arm-folded demeanour soon darkened as Cissé claimed his seventh goal in six games. This time Demba Ba’s attempt to play a slick one-two with Ben Arfa ended with Enrique challenging the Frenchman and the ball spinning clear to Cissé. The home No9, who had been clearly offside at the start of the move but was presumably then deemed inactive, proceeded to cut the ball back beyond Reina before finishing from close range.

Shortly afterwards Carroll stomped straight down the tunnel after being replaced by Dirk Kuyt with cries of “what a waste of money” ringing in his ears and then Reina was sent off.

Objecting to James Perch tripping him in the box, Liverpool’s goalkeeper reacted by barging into the Newcastle centre-half – deputising for the injured Fabricio Coloccini – with first his chest and then head. Having booked Perch, Atkinson then showed Reina a red card leaving Dalglish, who had used all three substitutes, to ask Enrique to take over in goal.